Ultrashort Laser Pulse Phenomena, 2nd Edition

568 Generation of Extreme Wavelengths

energy the electron can acquire before recombination, for which κ ≈ 3.2.

As a result, the emission spectrum has a sharp cutoff at the short wavelength side,

λ

min

≈ hc/



U

0

+ 3.2U

p

. Recollisions can occur with certain probability in each

half-cycle of the laser field, producing bursts of (attosecond) emissions coherent

to each other. In the frequency domain this corresponds to a comb of frequencies

representing the odd harmonics of ω



, the frequency of the driving field.

1

HH with

order n as large as 221 were observed [32]. With few cycle fundamental pulses,

only few emission bursts occur and the discrete (comb) structure of the emission

spectrum disappears. Extending the coherent continuum to the X-ray water

window (2.3 nm–4.4 nm) was accomplished with sub-10-fs pump pulses [33].

Single attosecond pulses were predicted theoretically for few-cycle funda-

mental pump pulses and demonstrated experimentally [34, 35]. This opened up

exciting new applications of high field physics and attosecond spectroscopy.

12.3. GENERATION OF ULTRASHORT

ACOUSTIC PULSES

An acoustic wave is a strain or shear wave that can be produced by piezo-

electric transducers in mechanical contact with the material. The transducers are

usually driven by rf voltages. Such techniques have gained importance for the

design of acousto-optical modulators for actively mode-locked lasers. Moreover,

acoustic pulse propagation, scattering, and reflection can conveniently be used

for material characterization.

The generation of acoustic waves with (fs) optical pulses is another example

that illustrates the complexity of processes associated with the interaction of

light pulses with matter. Let us assume that a fs pulse is incident on a solid

surface of a highly absorbing material and that its energy density is below the

threshold for plasma generation and other irreversible processes. Two effects

are responsible for launching an ultrasonic wave (pulse) into the material. Their

relative importance depends on the material properties and the parameters of

the (optical) excitation pulse (see, for example, Grahn et al. [36] and references

therein).

(a) If the pulse is absorbed in a semiconductor, a high-density distribution of

electron–hole pairs is created in a thin layer at the material surface. This

electron–hole plasma changes locally the effective potential that deter-

mines the arrangement of the atoms in the lattice. The resulting stress

is given by σ = N(dV

g

/dη) where N is the excited carrier density and

dV

g

/dη is the deformation potential.

1

For symmetry reasons (inversion symmetry) only odd harmonics are possible.

Generation of Ultrashort Acoustic Pulses 569

(b) The carriers excited to states above the band gap relax (cool) mainly

because of interaction with the lattice through electron–phonon collisions.

Consequently, the temperature of the excited volume rises, producing

an elastic stress. The stress amplitude is directly related to the thermal

expansion coefficient and the absorbed energy density. For a more detailed

analysis one has to account for a possible diffusion of the excited carriers

during the electron–phonon interaction. This may increase the effective

excited region.

Figure 12.10 illustrates a simple model of acoustic pulse formation. At t = 0

elastic stress is generated with a depth profile that follows approximately the

absorbed energy density. This stress distribution acts as source for a strain wave

(pulse) propagating into the material and toward the interface. On reflection at

the interface, the latter experiences a phase change of π. This gives rise to the

final shape of the strain pulse as shown in Fig. 12.10 for t = d/ν

ac

. According

to our model the width of the stress pulse is determined by the absorption length

z

a

for the optical pulse if the propagation length of the strain wave during the

optical excitation is smaller than z

a

.Ifν

ac

is the sound velocity this condition

can be expressed as

τ

p

<

z

a

ν

ac

. (12.7)

For an order of magnitude estimate let us assume z

a

≈ 10

−8

m and ν

ac

≈ 10

4

m/s,

which yields τ

p

< 1 ps. For optical pulses shorter than 1 ps, the duration of the

acoustic pulse depends mainly on material parameters rather than on the particular

0d

Distance from surface

0

Distance from surface

Stress (a.u.)

td/ν

ac

t0

0

0

Figure 12.10 Propagation of a stress pulse generated at t = 0 through absorption of an optical

pulse. (Adapted from Grahn et al. [36].)